Rational Solutions of Riccati-like Partial Differential Equations

Abstract

When factoring linear partial differential systems with a finite-dimensional solution space or analysing symmetries of nonlinear ODEs, we need to look for rational solutions of certain nonlinear PDEs. The nonlinear PDEs are called Riccati-like because they arise in a similar way as Riccati ODEs. In this paper we describe the structure of rational solutions of a Riccati-like system, and an algorithm for computing them. The algorithm is also applicable to finding all rational solutions of Lie's system {∂xu + u 2 + a 1 u + a 2 v + a 3 , ∂yu + uvwhere a 1 , . . . , d 3 are rational functions of x and y.

Rational general solutions of algebraic ordinary differential equations

“…Many other interesting results on finding Liouvillian solutions of linear ODEs were reported in [1,4,5,6,7,8,12,22,30,31,33,34]. In [21], Li and Schwarz gave the first method to find rational solutions for a class of partial differential equations.…”

Factoring zero-dimensional ideals of linear partial differential operators

“…Most of the results stated in this paper hold for several variables. We confine ourselves to the case of two variables, because a generalization of the algorithm in [7] for several variables is still on the way.…”

“…The notion and computation of differential Gröbner bases in K[∂x, ∂y] (see [5,3,9]) make sure that the system to be factored and the factors to be sought are of required dimensions. The algorithm in [7] enables us to compute firstorder factors. The idea of associated equations [1,10,11,2] inspires us to reduce our factorization problem to that of finding first-order factors.…”

Loewy decomposition of linear differential equations

“…In a series of publications Li, Schwarz and Tsarev [36][37][38] considered such systems of pde's in the plane and showed that a theory similar as for the ordinary case may be developed.…”